ABSTRACT A simulated annealing approach to the assignment of program tasks to processors in a distributed computer system is presented. Tasks of a program require certain capacitated computer resources. They also communicate at a given rate. Processors are interconnected by a communication network constituted of various types of links: local area network (LAN), wide area network (WAN) and specialised links. The communication resources are also capacitated. The purpose is to find the assignment of tasks to processors such that a measure of performance is optimised, the requirements of each task are met and the capacities of the resources are not violated. Various versions of the problem are identified and formulated. The design of the simulated annealing algorithm to solve the most general version is then described. The results of computational experience are reported

Surinder Kumar

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International Journal of Physics and Mathematical Sciences ISSN: 2277-2111 (Online) 
An Open Access, Online International Journal Available at http://www.cibtech.org/jpms.htm 
2014 Vol. 4 (2) April-June, pp. 39-44/Kumar 
Research Article 
© Copyright 2014 | Centre for Info Bio Technology (CIBTech)  39 
 
A SIMULATED ANNEALING APPROACH TO ASSIGNMENT PROBLEM 
OF PROGRAM TASKS TO PROCESSORS 
*Surinder Kumar 
Department of Mathematics, S.G.G.S College, Sec-26, Chandigarh, India 
*Author for Correspondence 
 
ABSTRACT  
A simulated annealing approach to the assignment of program tasks to processors in a distributed 
computer system is presented. Tasks of a program require certain capacitated computer resources. They 
also communicate at a given rate. Processors are interconnected by a communication network constituted 
of various types of links: local area network (LAN), wide area network (WAN) and specialised links. The 
communication resources are also capacitated. The purpose is to find the assignment of tasks to 
processors such that a measure of performance is optimised, the requirements of each task are met and the 
capacities of the resources are not violated. Various versions of the problem are identified and formulated. 
The design of the simulated annealing algorithm to solve the most general version is then described. The 
results of computational experience are reported. 
 
Keywords: Computer Networks, Execution Cost, Communication Cost, Task Assignment, Combinatorial 
Optimization, Distributed Computer System, Simulated Annealing 
 
INTRODUCTION 
Optimised assignment of program tasks to processors in a distributed system is an important problem, 
whose effective solution is useful during all phases of system development. During the design phase, it 
helps in determining the configuration that attains a certain performance level during the operational 
phase, it facilitates optimising the use of available resources; and during reconfiguration, it aids in 
reassigning tasks in response to system changes.  
The problem, in various versions, has received considerable attention in the literature on parallel and 
multi-processor configurations; see for example Berman and Snyder (1987); Billionet et al., (1992); 
Bokhari, (1987); Houstis and Aboelaze, (1991); Nicol and O’Halloran, (1991); and Norman and 
Thanisch, (1993). Existing algorithms fall into the following categories:  
 Exact algorithms applied to simplified problems (Fernandez-Baca and Medepalli, 1993; Sinclair, 1987). 
Heuristic algorithms applied to the full problem or a simplified version (Fernandez-Baca and Medepalli, 
1993; Hagin et al., 1996; Lo, 1384-1397).  
 Exact algorithms applied to the full problem (Hagin et al., 1996). 
However, since the problem is NP-complete, solving it exactly, for example by using branch and bound 
schemes (Hagin et al., 1996), is possible only for small problem instances. Therefore, employing 
simulated annealing, which has proved quite effective in solving to near optimality many combinatorial 
optimisation problems (Reeves, 1993), is justified.  
Problem Definition 
The tasks of a program have a computation time requirement and a memory requirement. They also, 
typically, communicate data with other tasks at given rates. Since computer and communication resources 
are capacitated, the problem of task assignment arises: finding the assignment of tasks to processors such 
that a measure of performance is optimised, the requirements of each task are met and the capacities of 
the resources are not violated. 
Communication requirements may be represented by a directed graph ),( EVG , called the communication 
graph (Figure 1), whose vertices V represent the modules and where the presence of an edge Eji ),(
signifies that communication occurs between tasks i and j. Each edge may be weighted by the channel 
capacity required for the communication to take place.  
International Journal of Physics and Mathematical Sciences ISSN: 2277-2111 (Online) 
An Open Access, Online International Journal Available at http://www.cibtech.org/jpms.htm 
2014 Vol. 4 (2) April-June, pp. 39-44/Kumar 
Research Article 
© Copyright 2014 | Centre for Info Bio Technology (CIBTech)  40 
 
Depending on context, several versions of the problem may be formulated. To do so, define the following 
parameters: 
:im  memory requirements of task i. 
:ip  processing requirements of task i. 
:nM  memory capacity of processor n. 
:nP  processing capacity of processor n. 
:ijnlc  communication cost between task i and j if i is assigned to processor n and j is  
 assigned to processor l. 
:inc  cost of executing task i on processor n. 
:ijd  communication load between tasks i and j. 
:nlD  capacity of the virtual communication link between processor n and l. 
 
 
Figure 1: An example of the communiucation requirement graph 
 
and the variable:  







,0
,1
otherwise
nprocessortoassignedisitaskif
xin   
Constrained module allocation: This problem version was motivated by system where a host computer 
with a large memory shares its load with a smaller, more specialised processor with limited memory 
capacity (Rao et al., 1979). Thus the problem is to minimise total system cost, subject to a resource 
constraint on the smaller processor, for convenience assumed to be processor 1. It can be formulated as: 
)3(
)2(,1
)1(min
1
),(
Mxm
ix
tosubject
xxcxc
i
ii
n
in
Eji n l
jlinijnl
i n
inin





 

 
Balanced module allocation: Here the objective is to minimise the maximum processor load, as a way of 
obtaining a balanced assignment (Chu and Lan, 1987). The total load on a processor for a given 
assignment is the total execution cost of the tasks assigned to it plus the sum of the communication costs 
from that task to all other tasks. It is worth noting that if two communicating tasks are assigned to 
International Journal of Physics and Mathematical Sciences ISSN: 2277-2111 (Online) 
An Open Access, Online International Journal Available at http://www.cibtech.org/jpms.htm 
2014 Vol. 4 (2) April-June, pp. 39-44/Kumar 
Research Article 
© Copyright 2014 | Centre for Info Bio Technology (CIBTech)  41 
 
different processors then the communication cost contributes to the load on both processors. The problem 
can be formulated as: 
)6(
)5(,1
)4(maxmin
),(
nMxm
ix
tosubject
xxcxc
n
i
in
n
in
Eji l
jlinijnl
i
inin
n
i











 

 
General constrained task allocation. The objective here is to minimise total cost subject to both memory 
and processing capacity constraints for each processor. The problem can be formulated as (1) Subject to 
)9(,
)8(,
)7(,1
nPxp
nMxm
ix
nin
i
i
n
i
ini
n
in






 
General constrained task allocation with network constraints. In this problem, it is required to take 
account of the limited capacities of the resources that constitute the communication network (LANs, 
WANs and direct links) as shown in Figure 2. 
One approach is to define a virtual link between each pair of processors and assign to it an appropriate 
capacity. Thereafter, it is assumed that if a pair of tasks is assigned to the pair of processors 
 
 
Figure 2: An example of a communication network 
 
under consideration, then the tasks impose a known communication load on the virtual link between the 
processors. The problem can then be formulated as (1) subject to (7)-(9) and 
)10(.,lnDxxd
i j
nljlinij   
A more flexible approach is to identify the possible communication paths between each pair of processors 
and the communication resources on which each path is incident. The problem then becomes that of 
assigning tasks to processors, as well as choosing the communication path between each pair of 
processors such that the capacity of each communication resource is not violated. The cost of 
International Journal of Physics and Mathematical Sciences ISSN: 2277-2111 (Online) 
An Open Access, Online International Journal Available at http://www.cibtech.org/jpms.htm 
2014 Vol. 4 (2) April-June, pp. 39-44/Kumar 
Research Article 
© Copyright 2014 | Centre for Info Bio Technology (CIBTech)  42 
 
communication between each specific two tasks assigned to each two specific processor could also be 
made dependent on the communication path chosen. By introducing binary variables to indicate the 
choice of each path or otherwise, the problem can be cast in a formulation similar to the one just cited 
above. However, such a formulation is cumbersome and is left out for that reason. 
It is this last version of the problem that will be addressed in the sequel. However, it will become clear 
that the approach presented is applicable equally well to the less general versions. 
)13(,1
)12(
)11(,



jlinijnl
jlijnl
inijnl
xx
x
x



 
The above formulations are all binary, nonlinear programming problems. It is possible to transform them 
into binary linear programming problems using standard device. Thus let ijnl  be a binary variable such that 
then each occurrence of jlinxx can be substituted by ijnl ,while adding the appropriate constraints (11)-
(13). However, it is clear that this is at the expense of a significant increase in the number of both 
variables and constraints.  
Whatever the version and whether a non-linear or a linear formulation is adopted, clearly, attempting to 
solve the task allocation problem to optimality is bound to fail for all bar insignificantly small problem 
instances. 
The Simulated Annealing Algorithm 
Since the basic outline of the simulated annealing algorithm is well known, we will confine ourselves 
here to presenting our development of its basic characteristic component; namely, the coding of solutions, 
the neighbourhood structure and the cooling schedule, for the problem under consideration.  
Coding of solutions. Solutions are coded by assigning a program task to a processor and choosing a 
communication path between each pair of processors. Let the number of processors be N, the number of 
paths between processors i and j be ijP and the number of tasks be M. Two array variables, )1( My and
)( NNz  , are then defined such that:  
 ),...,2,1( Nym  ; Where my  is the number of the processor to which task m is assigned,  
 ),...,2,1( ijij Pz  , ),...,2,1(, Nji  ; where ijz is the number of the path used for communication 
between processors i and j. 
Neighbourhood structure: A neighbouring solution is obtained by choosing at random one of two moves. 
The first is effected by moving a by choosing at random a task m from one processor to another (thus 
changing the value of ijy ), while the second is carried out by substituting at random the communication 
path between two processors thus randomly changing the value of the chosen variable ijz .The choice of 
type move is done with equal probability. 
Calculating the cost. Tables of costs and resource utilisation coefficients are assembled a priori, so that 
the cost of moves can be calculated incrementally, independently of problem size. Two matrices 
containing the sub-costs of communication and assignment are kept from one iteration to another and the 
change is applied to modified variables only. Constraint violations are handled by adding appropriate 
penalties.  
Cooling schedule. A geometric cooling schedule is used. The temperature is reduced in so that
kk TT  1 , where   is a constant less than 1. The chain at each temperature is updated in a similar 
manner, kk LL  1 , where   is a constant greater than1. 
International Journal of Physics and Mathematical Sciences ISSN: 2277-2111 (Online) 
An Open Access, Online International Journal Available at http://www.cibtech.org/jpms.htm 
2014 Vol. 4 (2) April-June, pp. 39-44/Kumar 
Research Article 
© Copyright 2014 | Centre for Info Bio Technology (CIBTech)  43 
 
The initial temperature is set after executing a sufficiently large number of random moves, such that the 
worst move would be allowed. 
Given  
 1m and 2m ,m the numbers corresponding cost reduction cost increase,  
 respectively. 
  : the average cost increase value of the 2m trials. 
 T : the initial temperature,  
 a : the desired initial acceptance value. 
The following relation may then be written: 
 
  

















aa
m
m
T
giveswhich
mm
emm
a
T
1log
,
2
1
21
/
21


 
The final temperature is chosen for to give a low acceptance value. 
 
Table 1: Computation times for five various size case studies 
Case No. of 
modules 
No. of 
processors 
No. of paths b/w two 
processors 
Total no. of 
variables 
Computation 
time (min) 
1 7 5 3 110 14 
2 9 6 3 162 24 
3 10 7 3 217 37 
4 12 8 3 288 54 
5 14 10 3 440 103 
 
RESULTS AND CONCLUSIONS 
The algorithm has been coded in Matlab, with the computation time optimized by using the parallel 
computation options. Several real-life problem instances have been solved satisfactorily. The computation 
times proved to be rather long. For example, a problem instance involving 10 processors, 7 sub networks, 
14 program tasks and 14 information flow edges required 103 minutes on an HP 712/60 workstation (for 
other cases refer to Table 1). However, computation time could be reduced drastically if the algorithm is 
coded and run in a more efficient environment; for example in C. 
Although the solutions obtained appear excellent, there are no benchmark problems with known solutions 
available. For this reason, among others, we are currently developing a branch and bound algorithm with 
a linear evaluation procedure, in the hope of solving at least medium size problem instances for 
comparison. 
 
REFERENCES 
Berman F and Snyder L (1987). On mapping parallel algorithms in parallel architecture. Journal of 
Parallel and Distributed Computing 4(5) 439-458. 
Billionet A, Costa MC and Sutter A (1992). An efficient algorithm for a task allocation problem. 
Journal of ACM 39 502-518. 
Bokhari S (1987). Assignment Problems in Parallel and Distributed Computing (Kluwer, Boston). 
Chu WW and Lan LM (1987). Task allocation and precedence relations for distributed real-time 
systems. IEEE Transactions on Computers 36 667-669. 
International Journal of Physics and Mathematical Sciences ISSN: 2277-2111 (Online) 
An Open Access, Online International Journal Available at http://www.cibtech.org/jpms.htm 
2014 Vol. 4 (2) April-June, pp. 39-44/Kumar 
Research Article 
© Copyright 2014 | Centre for Info Bio Technology (CIBTech)  44 
 
Fernandez-Baca D and Medepalli A (1993). Parametric module allocation on partial k-trees. IEEE 
Transactions on Computers 42 738-742. 
Hagin A, Dermler G and Rotherml K (1996). Problem formulation, model and algorithms for mapping 
distributed multimedia applications to distributed computer systems, Technical Report, University at 
Stuttgart, Fakultat Informatik. 
Houstis CE and Aboelaze M (1991). A comparative performance analysis of mapping applications to 
parallel multi-processor systems: A case study. Journal of Parallel and Distributed Computing 17-29. 
Lo VM (1988). Heuristic algorithms for task assignment in distributed systems. IEEE Transactions on 
Computers 1384-1397. 
Nicol DM and O'Halloran DR (1991). Improved algorithms for mapping pipelined and parallel 
computations. IEEE Transactions on Computers 40 295-306. 
Norman MG and Thanisch P (1993). Models of machines and computation for mapping in multi-
computers. ACM Computing Surveys 3 163-302. 
Rao GS, Stone HS and Hu TC (1979). Assignment of tasks in a distributed processing system with 
limited memory. IEEE Transactions on Computers 28 291-299. 
Reeves CR (1993). Modern Heuristic Techniques for Combinatorial Problems (Blackwell Scientific 
Publications, Oxford). 
Sinclair JB (1987). Efficient computation of optimal assignments for distributed tasks. Journal of 
Parallel and Distributed Computing 4 342-362. 


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